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Surface integrals

  1. Jan 29, 2016 #1
    1. The problem statement, all variables and given/known data
    Evaluate integral A.n dS for A=(y,2x,-z) and S is the surface of the plane 2x+y=6 in the first octant of the plane cut off by z=4

    2. Relevant equations
    Integral A.n dS

    3. The attempt at a solution
    The normal to the plane is (2,1,0) so the unit normal vector is 1/sqrt3 (2,1,0).
    A.n= 1/sqrt3(2y,2x,0)

    So this is where I get stuck. I think it goes as follows....

    ds= dxdy

    => double integral 2/sqrt3 y + 2/sqrt3 x dxdy

    And I'm not sure how to find the limits....
     
  2. jcsd
  3. Jan 29, 2016 #2

    HallsofIvy

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    The plane is in the first octant so bounded by x= y= z= 0. Since there is no "z" in the equation of the plane, the plane is parallel to the z- axis so we need another z bound: we are told that the plane is bounded by z= 4. The line 2x+ y= 6 crosses the x-axis at (0, 6) and the y axis at (3, 0). We can take x going from 0 to 3 and, for each x, y from 0 to 6- 2x. Equivalently, we can take y going from 0 to 6 and, for each y, x= (6- y)/2. In either case, z goes from 0 to 4.
     
  4. Jan 29, 2016 #3
    Would there not only be two bounds as its a surface integral? Forgive me if I'm being thick....
     
  5. Jan 29, 2016 #4

    HallsofIvy

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    Yes, the bounds on x and y. I didn't say you had to use the z bounds!
     
  6. Jan 29, 2016 #5

    LCKurtz

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    I would add that the problem is incompletely stated without the orientation of the surface being given.
     
  7. Jan 29, 2016 #6

    LCKurtz

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    Upon looking more closely at this problem and proposed solution -- you cannot paramaterize this surface in terms of the variables ##x## and ##y## because they are dependent on each other. I would suggest the parameterization ##x = x,~y=6-2x,~z = z##. So the surface is parameterized as ##\vec R(x,z) = \langle x, 6-2x, z\rangle## and use the formula$$
    \iint_S \vec A\cdot d\vec S = \pm\int\int \vec A \cdot \vec R_x\times \vec R_z~dxdz$$where the limits are over the ##xz## rectangle and the sign is chosen for the correct orientation (which needs to be specified).
     
  8. Mar 12, 2016 #7
    pardon me. i faced a similar prblm . could any1 pls xplain what is first octant??
     
  9. Mar 12, 2016 #8

    HallsofIvy

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    The "first octant" is the octant in an xyz- coordinate system in which all coordinates are positive.
    The "first octant of the plane" doesn't make sense. A plane has four quadrants not octants. I presume that you mean the portion in the first octant of the xyz coordinate, x> 0, y> 0, 2x+ y< 6 and 0< z< 4.

    2. Relevant equations
    Integral A.n dS

    3. The attempt at a solution
    The normal to the plane is (2,1,0) so the unit normal vector is 1/sqrt3 (2,1,0).[/quote]
    No, the unit normal is 1/sqrt(5)(2, 1, 0).

    Since dS includes the factor sqrt(5) (not sqrt(3)) itself, there is really no point in using "n" at all. I would write, rather, [itex]\vec{A}\cdot \vec{dS}[/itex] where [itex]\vec{dS}= (2, 1, 0)dxdy[/itex] so that the integral will be [itex]\int_{x= 0}^3 \int_{y= 0}^{6- 2x} 2y+ 2x dydx[/itex]

    In the x,y plane, the line 2x+ y= 6 crosses the x-axis at (3, 0) and crosses the y-axis at (0, 6). The gives a triangle with vertices at (0, 0), (3, 0), and (0, 6) in the first quadrant of the xy- plane. You can take x going from 0 to 3 and, for each x, y from 0 to 6- 2x.
     
  10. Mar 12, 2016 #9
    thnx
     
  11. Mar 12, 2016 #10

    Ray Vickson

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    In your expression ##\vec{A} \cdot \vec{n} \, dS##, ##dS## is the surface area-element in the plane ##2x+y=6##. Thus ##dS = ds \, dz,## where is ##ds = \sqrt{(dx)^2 + (dy)^2}## is the length-element along the surface perpendicular to the ##z##-axis.
     
  12. Mar 12, 2016 #11

    LCKurtz

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    This is not correct. As I explained in post #6, this cannot be worked as an xy domain problem.
     
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